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Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form. (English) Zbl 1079.15005
The author introduces two canonical forms for Leonard pairs, the TD-D canonical form and the LB-UB canonical form. He gives several applications of his theory. As he proceeds through the paper he illustrates his results using two running examples which involve specific parameter arrays. Near the end of the paper he discusses how Leonard pairs correspond to the q-Racah polynomials and some related polynomials in the Askey scheme. At the end of the paper he presents some open problems concerning Leonard pairs.
The paper is divided into twenty eight sections: Introduction; Leonard systems; The relatives of a Leonard system; Leonard pairs and Leonard systems; Isomorphisms of Leonard pairs and Leonard systems; The adjacency relations; The eigenvalue sequences; The split sequences; A classification of Leonard systems; The notion of a parameter array; Parameter arrays and Leonard systems; The parameter arrays of a Leonard pair; The LB-UB canonical form, preliminaries; The LB-UB canonical form for Leonard systems; The LB-UB canonical form for Leonard pairs; How to recognize a Leonard pair in LB-UB canonical form; Leonard pairs A, A* with A lower bidiagonal and A* upper bidiagonal; Examples of Leonard pairs A, A* with A lower bidiagonal and A* upper bidiagonal; The TD-D canonical form, preliminaries; The TD-D canonical map; The TD-D canonical form for Leonard systems; The TD-D canonical form for Leonard pairs; How to recognize a Leonard pair in TD-D canonical form; Examples of Leonard pairs in TD-D canonical form; Leonard pairs A, A* with A tridiagonal and A* diagonal; How to compute the parameter arrays; Transition matrices and polynomials; Directions for further research.

##### MSC:
 15A04 Linear transformations, semilinear transformations 15A21 Canonical forms, reductions, classification
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##### References:
 [1] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019 [2] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Berlin · Zbl 0747.05073 [3] Caughman, J.S., The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete math., 196, 1-3, 65-95, (1999) · Zbl 0924.05067 [4] Curtin, B.; Nomura, K., Distance-regular graphs related to the quantum enveloping algebra of $$\mathit{sl}(2)$$, J. algebraic combin., 12, 1, 25-36, (2000) · Zbl 0967.05067 [5] Curtin, B., Distance-regular graphs which support a spin model are thin, 16th british combinatorial conference, London, 1997, Discrete math., 197/198, 205-216, (1999) · Zbl 0929.05095 [6] Go, J., The Terwilliger algebra of the hypercube, European J. combin., 23, 4, 399-429, (2002) · Zbl 0997.05097 [7] Granovskiĭ, Ya.I.; Zhedanov, A.S., Nature of the symmetry group of the 6j-symbol, Zh. eksper. teoret. fiz., 94, 10, 49-54, (1988) [8] Granovskiĭ, Ya.I.; Lutzenko, I.M.; Zhedanov, A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. physics, 217, 1, 1-20, (1992) · Zbl 0875.17002 [9] Granovskiĭ, Ya.I.; Zhedanov, A.S., Linear covariance algebra for $$\mathit{sl}_q(2)$$, J. phys. A, 26, 7, L357-L359, (1993) · Zbl 0784.17019 [10] Grünbaum, F.A., Some bispectral musings, (), 31-45 · Zbl 0944.34062 [11] Grünbaum, F.A.; Haine, L., The q-version of a theorem of Bochner, J. comput. appl. math., 68, 1-2, 103-114, (1996) · Zbl 0865.33012 [12] Grünbaum, F.A.; Haine, L., Some functions that generalize the askey – wilson polynomials, Comm. math. phys., 184, 1, 173-202, (1997) · Zbl 0871.33009 [13] Grünbaum, F.A.; Haine, L., On a q-analogue of the string equation and a generalization of the classical orthogonal polynomials, (), 171-181 · Zbl 0938.33010 [14] Grünbaum, F.A.; Haine, L., The Wilson bispectral involution: some elementary examples, (), 353-369 · Zbl 0924.34074 [15] Grünbaum, F.A.; Haine, L.; Horozov, E., Some functions that generalize the krall – laguerre polynomials, J. comput. appl. math., 106, 2, 271-297, (1999) · Zbl 0926.33007 [16] Ito, T.; Tanabe, K.; Terwilliger, P., Some algebra related to P- and Q-polynomial association schemes, (), 167-192 · Zbl 0995.05148 [17] Koekoek, R.; Swarttouw, R.F., The Askey scheme of hypergeometric orthogonal polynomials and its q-analog, (1998), Report 98-17, Delft University of Technology, The Netherlands. Available at [18] Koelink, H.T., Askey – wilson polynomials and the quantum $$\mathit{su}(2)$$ group: survey and applications, Acta appl. math., 44, 3, 295-352, (1996) · Zbl 0865.33013 [19] Koelink, H.T., q-krawtchouk polynomials as spherical functions on the Hecke algebra of type B, Trans. amer. math. soc., 352, 10, 4789-4813, (2000) · Zbl 0957.33014 [20] Koelink, H.T.; Van Der Jeugt, J., Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. math. anal., 29, 3, 794-822, (1998), (electronic) · Zbl 0977.33013 [21] Koelink, H.T.; Van der Jeugt, J., Bilinear generating functions for orthogonal polynomials, Constr. approx., 15, 4, 481-497, (1999) · Zbl 0941.33011 [22] Koornwinder, T.H., Askey – wilson polynomials as zonal spherical functions on the $$\mathit{su}(2)$$ quantum group, SIAM J. math. anal., 24, 3, 795-813, (1993) · Zbl 0799.33015 [23] Leonard, D.A., Orthogonal polynomials, duality, and association schemes, SIAM J. math. anal., 13, 4, 656-663, (1982) · Zbl 0495.33006 [24] Leonard, D.A., Parameters of association schemes that are both P- and Q-polynomial, J. combin. theory ser. A, 36, 3, 355-363, (1984) · Zbl 0533.05016 [25] H. Rosengren, Multivariable orthogonal polynomials as coupling coefficients for Lie and quantum algebra representations, Centre for Mathematical Sciences, Lund University, Sweden, 1999 · Zbl 0946.33013 [26] Rotman, J.J., Advanced modern algebra, (2002), Prentice Hall Saddle River, NJ · Zbl 0997.00001 [27] Terwilliger, P., Introduction to leonard pairs and leonard systems, Algebraic Combinatorics, Kyoto, 1999, Sūrikaisekikenkyūsho Kōkyūroku, 1109, 67-79, (1999) · Zbl 0957.15500 [28] Terwilliger, P., Introduction to leonard pairs, OPSFA, Rome, 2001, J. comput. appl. math., 153, 2, 463-475, (2003) · Zbl 1035.05103 [29] Terwilliger, P., Leonard pairs from 24 points of view, Rocky mountain J. math., 32, 2, 827-887, (2002) · Zbl 1040.05030 [30] Terwilliger, P., A characterization of P- and Q-polynomial association schemes, J. combin. theory ser. A, 45, 1, 8-26, (1987) · Zbl 0663.05016 [31] Terwilliger, P., The subconstituent algebra of an association scheme I, J. algebraic combin., 1, 4, 363-388, (1992) · Zbl 0785.05089 [32] Terwilliger, P., The subconstituent algebra of an association scheme II, J. algebraic combin., 2, 1, 73-103, (1993) · Zbl 0785.05090 [33] Terwilliger, P., The subconstituent algebra of an association scheme III, J. algebraic combin., 2, 2, 177-210, (1993) · Zbl 0785.05091 [34] Terwilliger, P., A new inequality for distance-regular graphs, Discrete math., 137, 1-3, 319-332, (1995) · Zbl 0814.05074 [35] Terwilliger, P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear algebra appl., 330, 149-203, (2001) · Zbl 0980.05054 [36] Terwilliger, P., Two relations that generalize the q-Serre relations and the dolan – grady relations, (), 377-398 · Zbl 1061.16033 [37] Zhedanov, A.S., “hidden symmetry” of askey – wilson polynomials, Teoret. mat. fiz., 89, 2, 190-204, (1991) · Zbl 0744.33009
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